The parameters \(d_2^\star \) , \(d_{123}^\star \) , \(M_V\) , \(e_m^V\)  and the \(\eta -\eta ^{\prime }\)  mixing parameters are all fitted to experimental data from
BaBar ([1]} for \(\pi \) -TFF and [2]} for \(\eta ^{(\prime )}\) -TFF), CLEO [3]} and CELLO [4]},
LEP [5]} and Belle [6]}. The complete details of the fits can be seen on ref. [7]}. To compute \(a_\mu ^{P,HLbL}\) 
we rely on the loop integral representation of ref. [8]}. The \(a_\mu ^{P,HLbL}\)  contribution is obtained using a MonteCarlo run
based on a normal distribution for the fit parameters, including correlationsBy making the same calculation for the TFF in the chiral
limit, we obtain \(a_\mu ^{P,HLbL}=8.27\cdot 10^{-10}\) , where the change is essentially
given by the \(\eta \)  contribution. We note that, however, we keep the physical masses for the integration kernels.,
\(a_\mu ^{P,HLbL}= (8.47 \pm 0.16)\cdot 10^{-10}.\) 
